The war of words between Aspen Insurance Holdings Ltd. and Endurance Specialty Holdings Ltd. intensified late Monday after Endurance went public with an increased offer in its hostile pursuit of Aspen, an offer Aspen had previously rejected in private.

Endurance also announced that it would take its offer directly to Aspen shareholders and call a special meeting of shareholders under Bermuda law, where the companies are domiciled.

Aspen called Endurance's $2 bump to $49.50 per share “a backwards step,” according to Aspen Chairman Glyn Jones, speaking in a statement issued late Wednesday.

“Given Aspen's strong 4.4% book value growth in the first quarter, Endurance's new proposal represents an even lower multiple of book value per share than its initial proposal, and the stock portion of the proposal lags even further behind given the decline in Endurance's stock price since its initial proposal,” Mr. Jones said in the Aspen statement.

Mr. Jones also reiterated Aspen's dismissal of Endurance's logic for the deal.

“In addition to grossly undervaluing Aspen, the proposal represents a strategic mismatch and, based on our conversations with major clients and brokers, would result in significantly greater dis-synergies than Endurance claims,” Mr. Jones said in the statement.

Endurance fired back just hours later.

“The shrill rhetoric of Aspen's deeply entrenched board merely underscores the need to give Aspen shareholders the voice they deserve, which is exactly what our actions this morning are designed to do,” an Endurance spokesperson said in a statement.

The Endurance statement said that with less than 1.4% company ownership, Aspen's board and management are not “aligned” with shareholders.

“What Aspen's board and management have failed to achieve for 10 years, we are prepared to deliver today,” concluded the Endurance statement.

The exchange is the latest chapter in a drama which began with Endurance's April 14 $3.2 billion offer to buy Aspen, which was rejected after also having been made in private previously.